Peter F Erickson

Peter F. Erickson
Residence	Vancouver, WA, United States
Nationality	USA
Known for	The spatial infinitesimal, the veritable number system, absolute space and time, Infinity
Scientific career
Fields	Philosopher, Mathematician, Author

Peter F. Erickson is an American independent scholar and philosopher of mathematics who has spent much of his working life on a single question: whether space, time and number are infinitely divisible. His answer is that they are not. Against the orthodoxy of modern analysis, he holds that space is composed of indivisible infinitesimals — final elements that cannot be cut further — that time likewise consists of discrete instants, and that there is no "microscopic infinity" waiting below the smallest scale. From that starting point he defends absolute space, absolute time and absolute motion against Einstein, rejects non-Euclidean geometry as "a mere manipulation of symbols", argues that the limit concept in the calculus conceals a real problem rather than solving it, and claims the discovery of a third number system — the veritable number system — hidden inside the square root of negative one.

He presented twelve papers to the Natural Philosophy Alliance between 2005 and 2013, and has set out the full case in four books, principally Absolute Space, Absolute Time, & Absolute Motion (2006) and The Nature of Infinitesimals (2012).

Biography

Erickson graduated from Stanford University in 1961; according to his author biography he did so Phi Beta Kappa. He lives in Vancouver, Washington. He has held no academic post, and his work has been carried out and published independently of any institution.

His writing began outside physics altogether. His author biography records a 1975 work, Introduction to the Tripartite System, proposing a new monetary arrangement intended to forestall what he expected to be the eventual collapse of the dollar. In 1997 he published The Stance of Atlas: An Examination of the Philosophy of Ayn Rand, a critical study cast as a Platonic dialogue among four characters, with one — Philosophus — carrying the author's own voice. It argues that Rand held an unstable position between monism and pluralism, that important parts of her similarity-based theory of concepts are fallacious, and that her solution to the problem of universals fails; Erickson offers his own in its place. A book on the stock market, Passport to Poverty, followed in 2003.

The turn to mathematics and physics came next, and the first paper on the veritable number system was, by his account, copyrighted and circulated in 1999. From 2005 he was a regular presenter at Natural Philosophy Alliance conferences, appearing at nine consecutive meetings through 2013.

The spatial infinitesimal

Erickson's central doctrine is that space has a smallest part. The standard view — that any interval can be divided without end — he regards not as a proven fact but as an assumption that can be refuted, and one that has cost mathematics its grip on reality.

Space is three-dimensional only, and consists of infinitesimals. Infinitesimals are points of location, without area, shapeless, indivisible, continuous in all directions. Nothing inconsistent with them, such a square circle, can exist. There is no microscopic infinity. Knowing that they exist provides the answer to the mystery of irrational numbers, asymptotes, "infinite series," and much else.

— Peter F. Erickson, "The Spatial Infinitesimal is the Final, Indivisible Element of Space" (2008)

The claim is doing more work than it may appear. If space bottoms out in indivisible elements, then several long-standing puzzles change character rather than remaining mysteries. An irrational magnitude inside the unit becomes comprehensible as a definite count of infinitesimals rather than an unending decimal; division by zero becomes valid under statable conditions; and the question "is space curved?" acquires a determinate answer. As he put it in the 2007 lecture drawn from his book:

Common sense teaches that space is different from the objects within it. This is also the conclusion of sustained reason. The key to understanding space is its smallest division, the infinitesimal. The opposing thesis, namely that space is infinitely divisible can be refuted.

— Peter F. Erickson, "The Spatial Infinitesimal" (2007)

The insistence that space is distinct from what occupies it is not incidental. It is the same commitment that separates absolute space from the relational and geometrical conceptions that replaced it, and it is why the infinitesimal doctrine and the case against Einstein are, for Erickson, one argument rather than two.

The nature of time

Time receives a parallel treatment, and a more radical one. In "The Nature of Time" (2008) Erickson argues that time does not move or flow, is not a fourth dimension, and is entirely independent of space although both must exist. It too consists of infinitesimals — instants — and each instant is at once discrete ("it is here, then no more") and continuous, since there is no time that is not a time. He adds a claim about how we know it: our consciousness of time does not come through the senses at all but from memory, making it an innate idea.

The consequence for physics is direct. A time made of indivisible instants cannot be the fourth axis of a continuous manifold, and motion cannot be a limit process across intervals that do not exist; on Erickson's account motion occurs instantaneously, from one instant to the next.

Absolute space, absolute time, absolute motion

Erickson's 2006 book states the thesis in its title and defends it from the infinitesimal upward rather than from experiment downward — a strategy that distinguishes him from most of the wiki's other critics of relativity, who argue from Michelson–Morley, Sagnac or GPS data.

Absolute Space, Absolute Time, and Absolute Motion exist. These are shown to be facts through an investigation of the nature of infinitesimals. Knowledge of that nature also makes the irrational magnitudes within the unit comprehensible. The number line is shown to be cognitively superior to set theory; furthermore, non-Euclidean geometry is shown to be a mere manipulation of symbols and not an expression of a "parallel universe."

— Publisher's description, Absolute Space, Absolute Time, & Absolute Motion (2006)

Two further claims from the same work are worth drawing out. The first is that the foundation of science "is not some vague generality, but the exercise of reason as originating from the human sensorium" — a thoroughgoing empiricism about the origin of concepts, which is what licenses his repeated appeals to common sense as evidence rather than as prejudice. The second is that there is no difference in kind between mathematical reasoning and ordinary inductive reasoning. If that is right, then mathematics has no special warrant to overrule intuition about space, and a geometry that cannot be pictured is not a discovery about reality but a formalism.

The limit and the discarded terms

Erickson's objection to the differential calculus is narrow, technical and, unusually for this literature, constructive: he thinks something has been thrown away that could be used.

In the standard derivation of a derivative, terms of higher order are dropped as the increment goes to zero. Erickson argues that the limit concept "is wrong" as applied here — that it does not solve the problem that the derivative generally differs from dy/dx but conceals it. Since the derivative is an exact product rather than an approximation, the terms conventionally discarded must, on inspection, still be present. His suggestion is that these residual terms are not bookkeeping debris but a "hidden key to a level of reality beneath that of nano-technology", and that recovering them "may provide an avenue for the future advance of physical science."

The veritable number system

Main article: Veritable Number System

Erickson claims to have found a number system inside the square root of negative one, which he calls the veritable number system.

Most consider it to be either something transcendental or a mere direction to perform a certain operation. Einstein used it to help make plausible his idea of the fourth dimension. Actually, neither the idealists nor the logical positivists are correct. The −1 inside the radical sign is from a number system discovered by the author. It is called the "Veritable Number System." … It stands with the real number system and absolute numbers as one of the three ways to handle the concept of direction.

— Peter F. Erickson, "The Solution to the Mystery of the Square Root of Minus One" (2005)

The system's distinguishing feature is that positive and negative veritable numbers are incompatible for the basic operations — addition, subtraction, multiplication, division, roots and ratios — so that sign is not a property that can be freely combined away. Erickson argues that this makes the system markedly better than the reals at characterising paths, whether straight or curvilinear, and he developed it across five further papers: on quaternions and vectors (2009), where he judges that neither Hamilton's quaternions nor the Gibbs vector calculus that displaced them got the matter right; on division (2011), which he holds to be more versatile in the veritable system than in the reals; and on negative numbers generally (2010), later expanded into the book The Nature of Negative Numbers (2011).

Against set-theoretic continuity

The last of Erickson's papers, "Bertrand Russell and "Continuity"" (2013), attacks the definition of continuity at its root — the requirement that in a continuum there is no "next-to", no immediate successor to a point.

Central to establishment's concept of continuity is that there be no "next-to." It is integral to set theory and modern mathematics. Bertrand Russell noticed that it is in conflict with the common sense understanding of differential equations. Nonetheless, he accepted it. This led to a bizarre notion of a "physical object." Is there an alternative to this concept of continuity? Yes, there is.

— Peter F. Erickson, "Bertrand Russell and 'Continuity'" (2013)

The complaint is not that Russell was careless but that he saw the conflict and chose the formalism over the physics. On Erickson's alternative there is a next-to: each infinitesimal has neighbours, and continuity means only that no gap intervenes between them. This is why he holds the number line to be "cognitively superior to set theory" — the line is a picture of an actual structure, whereas the set-theoretic continuum is, on his account, a construction that has quietly detached itself from the space it was supposed to describe.

Precursor: de Bothezat

Erickson's 2012 paper "George De Bothezat's Teaching on the Infinitesimal" identifies a predecessor. George de Bothezat (1882–1940) was the Russian-American engineer who built the US Army's first helicopter, the quadrotor "flying octopus" of 1922, and who in later life turned to attacking relativity. His Back to Newton: A Challenge to Einstein's Theory of Relativity (1936) organised its case around what he called the three great cognitive issues — infinity, absolute time and absolute motion — the same triad Erickson would take up seventy years later. De Bothezat pressed his objections in person: Einstein rebutted him from the floor at a lecture in Princeton in 1935.

Related work on this wiki

Erickson's defence of absolute space arrives at a conclusion many others here reach by other routes, and the contrast is instructive. Where he argues downward from the structure of the continuum, other papers catalogued here argue upward from experiment and paradox — among them Evidence for Newtonian Absolute Space and Time, Experimental Detection of Absolute Space, From Relativistic Paradoxes to Absolute Space and Time Physics, Towards the Reinstatement of Absolute Space, and Some Possible Cosmological Implications and The Semantics of Absolute Space. Erickson is, so far as the wiki's holdings show, the only contributor who tries to settle the question from the philosophy of mathematics alone.

His position also runs against the grain of another strand here: nonstandard analysis, which recovers rigorous infinitesimals precisely by embedding them in the set-theoretic framework Erickson rejects. Papers such as Nonstandard Analysis Applied to Special and General Relativity: The Theory of Infinitesimal Light-Clocks reach infinitesimals by a route he would not accept.

Reception

Erickson's work has attracted almost no engagement outside the Natural Philosophy Alliance. No mathematician or philosopher appears to have published on the veritable number system, and there is no academic response to The Stance of Atlas from either side of the Objectivist literature.

Independent reviews of the books have been cautiously respectful without endorsing the mathematics. Kirkus called Absolute Space, Absolute Time, & Absolute Motion "a serious, rarefied attempt to construe reality outside the dominant paradigm", praising the progression from finite to infinite and the handling of division by zero while faulting the density of the prose. Reviewing The Nature of Negative Numbers, the same service found "the author's ideas and reasoning are compelling" but judged the veritable system an intellectual curiosity rather than a working tool, on the ground that mathematics and science rest on the ordinary laws of real numbers. That verdict — interesting, coherent, unused — is a fair summary of the reception to date.

Papers on this wiki

• 2013 – Bertrand Russell and "Continuity"
• 2012 – George De Bothezat's Teaching on the Infinitesimal (read in full)
• 2011 – Division in the Veritable Number System (read in full)
• 2010 – On Understanding Negative Numbers (read in full)
• 2009 – Further Discoveries About the Veritable Number System
• 2009 – Quaternions, Vectors, and Veritable Numbers
• 2008 – The Nature of Time: There Exists an Infinitesimal of Time Within Which Division is Impossible
• 2008 – The Spatial Infinitesimal: The Spatial Infinitesimal is the Final, Indivisible Element of Space
• 2007 – The Hidden Opportunities in the Derivative
• 2007 – The Spatial Infinitesimal
• 2006 – Absolute Space, Absolute Time, & Absolute Motion
• 2005 – The Solution to the Mystery of the Square Root of Minus One

Books

• The Nature of Infinitesimals (CreateSpace, first published 2012; later edition 2016) — ISBN 1479701823. Listing
• The Nature of Negative Numbers (2011) — ISBN 1463761600; reissued 2017.
• Absolute Space, Absolute Time, & Absolute Motion (Xlibris, 2006; paperback 2008) — ISBN 1599261170. Listing
• Passport to Poverty: The '90s Stock Market and What It Can Still Do To You (Xlibris, 2003) — ISBN 1413404014.
• The Stance of Atlas: An Examination of the Philosophy of Ayn Rand (Herakles Press, 1997) — ISBN 0965418308.
• Introduction to the Tripartite System: A Monetary Program for Americans (1975).

See also

• Infinity
• Georges A de Bothezat
• Isaac Newton
• Natural Philosophy Alliance